WOLFRAM

Wolfram Language & System Documentation Center

Complexes

represents the domain of complex numbers, as in x∈Complexes.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
Cite this Page

Complexes

Complexes

represents the domain of complex numbers, as in x∈Complexes.

Details

  • x∈Complexes evaluates immediately only if x is a numeric quantity.
  • Simplify[expr∈Complexes] can be used to try to determine whether an expression corresponds to a complex number.
  • The domain of real numbers is taken to be a subset of the domain of complex numbers.
  • Complexes is output in StandardForm or TraditionalForm as TemplateBox[{}, Complexes]. This typeset form can be input using comps.

Examples

open all close all

Basic Examples  (3)

is a complex number:

Wolfram Language code: Element[E ^ (I Sin[7]), Complexes]

Exponential of a complex number is a complex number:

Wolfram Language code: Simplify[Element[E ^ x, Complexes], Element[x, Complexes]]

Find complex numbers that make an inequality well defined and True:

Wolfram Language code: Reduce[x ^ 2 < 1, x, Complexes]

Scope  (2)

Specify that all variables should be considered complex, even if they appear in inequalities:

Wolfram Language code: Reduce[x ^ 4 ≤ 1, x, Complexes]

By default, Reduce considers all variables that appear in inequalities to be real:

Wolfram Language code: Reduce[x ^ 4 ≤ 1, x]

For every real number y there exists a complex number whose square is real and less than y:

Wolfram Language code: Resolve[Exists[x, x ^ 2 < y], Complexes]

By default, Resolve considers all variables that appear in inequalities to be real:

Wolfram Language code: Resolve[Exists[x, x ^ 2 < y]]

TraditionalForm of formatting:

Wolfram Language code: Complexes//TraditionalForm

Properties & Relations  (2)

Complexes contains Reals, Algebraics, Rationals, Integers, and Primes:

Wolfram Language code: Refine[Element[x, Complexes], Element[x, #]]& /@ {Reals, Algebraics, Rationals, Integers, Primes}

Infinite quantities are not considered part of the Complexes:

Wolfram Language code: {Element[Infinity, Complexes], Element[ComplexInfinity, Complexes]}
Wolfram Research (1999), Complexes, Wolfram Language function, https://reference.wolfram.com/language/ref/Complexes.html (updated 2017).

Text

Wolfram Research (1999), Complexes, Wolfram Language function, https://reference.wolfram.com/language/ref/Complexes.html (updated 2017).

CMS

Wolfram Language. 1999. "Complexes." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/Complexes.html.

APA

Wolfram Language. (1999). Complexes. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Complexes.html

BibTeX

@misc{reference.wolfram_2026_complexes, author="Wolfram Research", title="{Complexes}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Complexes.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_complexes, organization={Wolfram Research}, title={Complexes}, year={2017}, url={https://reference.wolfram.com/language/ref/Complexes.html}, note=[Accessed: 12-June-2026]}